Constructing levels in arrangements and higher order Voronoi diagrams

We give simple randomized incremental algorithms for computing the less than or equal to k-level in an arrangement of n lines in the plane or in an arrangement of n planes in R-3. The expected running time of our algorithms is O(nk + n alpha(n)logn) for the planar case and O(nk(2) + nlog(3)n) for the three-dimensional case. Both bounds are optimal unless k is very small. The algorithm generalizes to computing the less than or equal to k-level in an arrangement of discs or x-monotone Jordan curves in the plane. Our approach can also compute the k-level; this yields a randomized algorithm for computing the order-k Voronoi diagram of n points in the plane in expected time O(k(n - k) logn + nlog(3)n).